A Game Theoretical View on Efficiency Wage Theories

Transcription

1 MPRA Munich Personal RePEc Archive A Game Theoretical View on Efficiency Wage Theories Dennis Wesselbaum The Kiel Institute for the World Economy, Kiel University, EABCN October 2009 Online at MPRA Paper No , posted 20. October :37 UTC

2 A Game Theoretical View on Eciency Wage Theories * Dennis Wesselbaum The Kiel Institute for the World Economy, Kiel University, and EABCN October 20, 2009 Abstract The eciency wage theory developed by Akerlof (1982) assumes observability of eort and the ability of rm and worker to commit on their eort/wage decisions. We show that, from a game theoretical point of view, we have to understand the rm/worker relationship as a repeated Prisoner's dilemma. Therefore, cooperation is per se not a (subgame perfect) Nash equilibrium and hence the Akerlof (1982) theory is based upon an implicit assumption of cooperation, which can not be implemented w.l.o.g.. In addition, we nd that this approach is a special case of the Shapiro and Stiglitz (1984) approach and hence unify the two approaches. Keywords: Eciency Wage, Prisoner's Dilemma, Repeated Game, Subgame Perfect Nash Equilibrium. JEL classication: C72, C73, J41. * I would like to thank seminar participants at the Kiel Institute for highly valuable comments. denniswesselbaum@web.de 1

3 1 Introduction In order to explain equilibrium unemployment, nobel laureate George A. Akerlof developed the fair wage theory based on the sociological partial gift exchange approach (see Akerlof (1982)). This theory explains equilibrium unemployment by the fact that the wage is above the market clearing wage. The reason for this nonneoclassical phenomenon is the fact that rms tend to pay a higher wage in order to ensure that workers provide a desired amount of eort. In contrast to Akerlof's idea, Shapiro and Stiglitz (1984) suggested a dierent type of eciency wage model. While in Akerlof's model eort is observable 1 and measurable, Shapiro and Stiglitz assume that eort is not observable. Consistently, a worker has an incentive to shirk, viz. the worker might not provide eort. Under equilibrium full employment, this would urge the rm to pay a wage that is above the market clearing wage, in order to create some punishment for shirking. As a direct consequence, equilibrium full employment is not feasible, since the higher wage will decrease labor demand and cause (equilibrium) unemployment. We see that both theories work along the same dimension, namely increasing the wage above the market clearing wage, but with dierent causes. However, there are several challenges aicted with Akerlof's idea in a dynamic context. Especially, the assumption of commitable eort is crucial. Akerlof (1982) states that "Since output is easily observable, it is at least a bit surprising... that workers are not paid wages proportional to their outputs." From our point of view, the observation of each worker is hardly imaginable due to monitoring costs and the possible negative psychological eects on the workers motivation. Consistently, and because the rm/worker pair sets the wage/eort at the beginning of the period - since the rm maximizes prots and denes its plan at the beginning of every period, while the household maximizes utility at the beginning of the period -, the proportional wage is only a hypothetical construct. 2 Let us consider a dynamic version based upon the Akerlof approach. 3 The underlying decision process can be deconstructed in the following steps 1. A rm-worker match exists or is created, 2. Firm's solve their optimization problem, 3. Firm's set the optimal wage, 4. Worker's receive the wage oer and set their eort, 5. Production commences. We infer that after the rm and the worker have somehow matched, the rm maximizes its prots and determines the optimal wage/eort decision. 4 The worker 1 And to be more precisely, it is possible for workers to commit on their eort "gift". 2 We would like to emphasize that our analysis is independent from the fact whether output (or eort) is observable. The main point is the commitability of eort. 3 See e.g. Danthine and Donaldson (1990) or Danthine and Kurmann (2004) for the dynamic version of the Akerlof approach in a RBC, NKM context respectively. 4 The model has no endogenous hiring process. Furthermore, frictions are only related to labor demand. The rm solves its maximization problem i.a. subject to the eort function, known 2

4 receives the wage and provides the desired amount of eort. Akerlof motivates this i.a. with the sociological gift exchange, i.e. workers have "sentiments" for the rm. However, the other side of the coin is that whenever there is sentiment, there is also the possibility of being discouraged by rm's decisions. We nd that the implicit assumption of Akerlof (1982) of cooperation being a (subgame perfect) Nash equilibrium of the corresponding game does not hold w.l.o.g.. Furthermore, we establish conditions for the (subgame) perfectness of the cooperation strategy for the entire time path of the game. In addition, we nd that this approach is a special case of the Shapiro and Stiglitz (1984) approach and hence unify the two approaches and provide a game theoretical foundation of eciency wages. The paper proceeds as follows. In the next section, we review the basic concepts of repeated prisoner's dilemma and in section 3, we have a game theoretic view on the rm/worker relationship. In section 4 we will draw the conclusion. 2 The Repeated Prisoner's Dilemma We consider an innitely repeated Prisoner`s dilemma 5, i.e. the horizon of the game T is unknown to both players and they both expect the game to be played for a long period of time. Let us call our game B expressed by a simultaneous-move matrix. Furthermore, we assume that our game is time-independent and hence stationary. Let P be the set of players containing the nite set {1, 2,..., n}. Our game starts in period t = 0 and is played every period. We assume that every player has full information and that actions are revealed to all players before the next round. With this assumption, we enable players to condition their actions on the history of events up to point t. In order to avoid the problem of inniteness of player's payos, we introduce discounting of future payos. We understand this discounting as a measure of time preference. 6 In the following, we will discuss the general mathematical background for later purpose. Let U t,i denote the utility function of player i over the outcomes of B in period t. In addition, let φ (0, 1) be the time-independent discount rate. Consistently, if the payos are constant over B and t, we can write the stream of payos of player i as φ t U t,i. (1) t=0 to the rm. Therefore we have the additional assumption of full information or revelation of information. 5 As usual in DSGE models, we assume that both players are innitely lived. 6 Notice that in Akerlof (1982) only static problems are considered. We introduce discounting, because worker and rm can not be sure how long the game will continue. For instance, the worker might expect that in a recession the rm is more likely to cheat or that the separation probability increases. 3

5 The average discounted value of the payo stream is then given by (1 φ) φ t Γ t i = (1 φ t )ˆΓ i + φ tˇγi, (2) t=0 where Γ t i is the constant payo of player i for t periods and let ˆΓ i denote the constant payo for the rst t periods, while ˇΓ i is the dierent payo for the next t periods. 7 Furthermore, let s t i = (s0 i, s1 i,...) denote the history-dependent strategies. The strategy prole s is the n-tuple of indiviual strategies such that s = (s 1,..., s n ). Now, let us consider a repeated-game between players (i, j) with the two strategies C and N. A strategy for player i that ensures cooperation is: play C in the rst period and in every consecutive period, i player j always cooperated. However, play N, i player j played N in the precedent period. Let player i s repeated game strategy be s i = ( s 0 i, s1 i,...). Consistently, in period t after history ht8 s t i(h t C, i h t = (C, C) t, ) = N, otherwise. As an illustrative example, consider the following game presented in Table 1. Table 1: A Prisoner's Dilemma Example i, j C N C 1,1-1,2 N 2,-1 0,0 If both players stick to their cooperation strategy, the payo computation is straightforward, resulting 1. Now, let player i deviate from C in period t. For the rst t periods, she receives 1 and for period t, she receives 2, i.e. the payo from (C, N). In any consecutive period t > t, both players choose N, i.e. to not cooperate, and receive 0. 9 Using equation (2) yields that, (1 φ) t=0 φ t Γ t i = (1 φ t )ˆΓ i + φ t ( (1 φ)ˇγ i + φ Γ i ), (3) if player i receives ˇΓ i only for period t. Consistently, player i s payo from deviating is given by 1 φ t (2φ 1). Some algebra yields that this cheating strategy is not protable, as long as φ 1 2. We have shown that cooperation is a Nash equilibrium of the game, if the time preference parameter is above a certain endogenously determined threshold. 7 (1 φ) t=0 φt Γ t i = (1 φ) ( j 1 t=0 φt Γ t i + t=j φt Γ t i ) ( ˆΓi (1 φ = (1 φ) t ) 1 φ + ˇΓ ) i φ t = (1 φ 1 φ t )ˆΓ i + φ t ˇΓ i, see Ratli (1997). 8 We assume that for any prole a it holds h 0 = (a) 0, such that h t = (C, C) t. 9 Here, we assume that the punishment for deviating is playing N for any consecutive period. One might assume dierent punishment strategies, but since we consider a rm/worker relationship cheating should result in separation and hence there should be no way back to rebuild the relationship. 4

6 As a nal step, we show that cooperation is a subgame perfect Nash equilibrium. For this purpose, consider a subgame that starts in period ˆt with history hˆt. The restriction ŝ to the subgame hˆt denes the strategy in this subgame. The restriction to this subgame is given by ŝ t i(hˆt C, i hˆt = (C, C)ˆt, ) = N, otherwise. Now, we can identify two classes of histories (i) both players have choosen to cooperate for the entire game, and (ii) at least one player cheated in at least one of the previous periods. Then, for class (i) subgames the restriction reduces to the game strategy derived in (3), because the history up to this subgame has to read as h t = hˆt = (C, C) t = (C, C)ˆt. Since in (3) s t i is a Nash equilibrium, the restriction ŝt i is a Nash equilibrium strategy prole in (i), i the condition for the discount factor holds. In the second class, and by assumption, because one player has choosen to noncooperate, both players choose to non-cooperate in the ongoing subgame. It can be shown, that such a strategy is also a subgame perfect equilibrium and consistently, that for any subgame the restriction of s t i is a Nash equilibrium for that subgame, i φ 1 2. Therefore, st i is also a subgame perfect Nash equilibrium of the repeated game. 3 The Eciency Wage Pendant We have shown that in a repeated Prisoner's dilemma cooperation is a (subgame perfect) Nash equilibrium, i the discount rate is above a certain threshold. In the following, we have to show that the eciency wage theory - from a game theoretic viewpoint - yields such a Prisoner's dilemma. Which variables do we have to consider? The payos contain the worker's eort e > 0, since the utility function in Akerlof (1982) has the form u(e n, e, w, ɛ), where ɛ represents the worker's taste and e n are the norms of eort. Moreover, eort is a decision variable for the worker and disregarding eort would lead to a distortion of our results. In addition, we consider the wage w > 0 - also present in the worker's utility function - to be some additional payment in case of the achievement of predetermined goals. 10 For the rm, only output y is considered. We assume that w > e, i.e. that the eort - given in terms of its share of the wage - is always smaller than the wage. Similarly, we assume y > w, such that the production process generates prots. Using these variables yields the following game-matrix presented in Table Those goals are written down in the contract between rm and worker and can be considered as performance pay (see Lemieux et al. (2007)). If we would include unemployment benets, w b would have to hold. 5

7 Table 2: The Eciency Wage Game F/W C N C (y-w),w-e -w,w N y,-e 0,0 If rm and worker cooperate, the rm receives the output and has to pay the "extra" wage, while the worker receives the wage and provides eort (which can not be used for leisure). If the rm cooperates, but the worker chooses to non-cooperate, the rm has to pay the wage (since it committed on paying the wage) but receives no output. 11 The same considerations hold vice versa for the case (N, C). If both players choose to non-cooperate, the rm receives no output and the worker can spend this eort for leisure, hence generating some amount of utility. If we now apply the methodology introduced in the precedent section, we infer that this is indeed a Prisoner's dilemma, since the static (subgame perfect) Nash equilibrium is (N, N), while 0 < (y w) and 0 < w e. Therefore, we can set up two propositions, such that cooperation is a (subgame perfect) Nash equilibrium in the innitely repeated game, i.e. that rm and worker choose to cooperate over the entire game. Proposition 1 The rm will cooperate, i y w φ. (4) Proof See the Appendix. Proposition 2 The worker will cooperate, i w e φ. (5) Proof See the Appendix. We have shown that the eciency wage theory - from a game theoretic view - is in fact a Prisoner's dilemma and, consistently, we established conditions, such that cooperation is a (subgame perfect) Nash equilibrium. 11 For the sake of simplicity, we assume an extreme case. However, this assumption leaves our qualitative results unaected. 6

8 4 Final Remarks We have shown that from a game theoretic viewpoint, the Akerlof approach is based upon an implicit assumption of commitability of eort. However, since the game between rm and worker is a Prisoner's dilemma, cooperation is per se no (subgame perfect) Nash equilibrium. We develop conditions for which cooperation is in fact a Nash equilibrium and consistently show that the Akerlof (1982) eciency wage theory is nested within the Shapiro and Stiglitz (1984) theory. Moreover, we can understand the former approach as a special case of the general game between rm and worker, i.e. the latter. With this game theoretic approach, we are able to unify these two ideas, often viewed as disparate. 7

9 Appendix Proof of Proposition 1 The rm will cooperate, i y w φ. (6) Consider the game presented in Table 2. If we apply (3) to this problem, we initially obtain (1 φ t )(y w) + φ t ((1 φ)y). (7) Consistently, the prot of cheating is given by y w + φ t (w φy). (8) However, if cheating should not be protable, the stream of prots has to be smaller than the prot from cooperation, i.e. y w + φ t (w φy) y w. (9) Applying some algebra yields the condition y w φ, (10) q.e.d. Proof of Proposition 2 The worker will cooperate, i w e φ. (11) The problem for the worker is solved analogously to the rms problem. Therefore, the initial condition looks as follows (1 φ t )(w e) + φ t ((1 φ)w). (12) Some rearranging gives w e + φ t e φ t+1 w. (13) The condition for non-protability of cheating is given by w e + φ t e φ t+1 w w e, (14) 8

10 such that w e φ. (15) q.e.d. References Akerlof, George A "Labor Contracts as Partial Gift Exchange." Quarterly Journals of Economics, (97(4): Danthine, Jean-Pierre and Andre Kurmann "Fair Wages in a New Keynesian Model of the Business Cycle." Review of Economic Dynamics, 7(1): Danthine, Jean-Pierre and John B. Donaldson "Eciency wages and the business cycle puzzle." European Economic Review, 34(7): Lemieux, Thomas, W. Bentley MacLeod, and Daniel Parent "Performance Pay and Wage Inequality." NBER Working Paper, No Ratli, Jim Lecture Slides to the Graduate-Level Course on Game Theory. Available via the following link: Shapiro, Carl and Joseph E. Stiglitz "Equilibrium Unemployment as a Worker Discipline Device." American Economic Review, 74(3):